Three immiscible transparent liquids with uniform refractive indices $\frac{3}{2}, \frac{4}{3}$ and $\frac{6}{5}$ are arranged one on top of another. The depths of the liquids are $3 \ cm, 4 \ cm$ and $6 \ cm$ respectively. The apparent depth of the vessel is (in $cm$)

The figure shows a ray of light entering and passing through a dense glass slab and emerging from the other side. If the angle of incidence $i=60^{\circ}$, slab thickness $b=0.04 \text{ m}$, and the refractive index of glass $\mu=\sqrt{3}$, the parallel shift $d$ between the emerging and entering rays in $\text{mm}$ is:

$A$ vessel is filled with two different liquids which do not mix. One liquid is $40 \ cm$ deep with a refractive index of $1.6$,and the other is $30 \ cm$ deep with a refractive index of $1.5$. What is the apparent depth of the vessel when viewed normally (in $cm$)?

$A$ glass beaker is filled with water up to $5 \,cm$. It is kept on top of a $2 \,cm$ thick glass slab. When a coin at the bottom of the glass slab is viewed at normal incidence from above the beaker,its apparent depth from the water surface is $d \,cm$. The value of $d$ is close to ........ $cm$ (the refractive indices of water and glass are $1.33$ and $1.5$,respectively).

$A$ fish at a depth of $12 \, cm$ in water is viewed by an observer on the bank of a lake. To what height the image of the fish is raised? $....... \, cm$ (Refractive index of lake water $= 4/3$)

The figure shows a transparent slab of length $1 \, m$ placed in air,whose refractive index in the $x$-direction varies as $\mu = 1 + x^2$ for $0 \leq x \leq 1$. The optical path length of ray $R$ will be: